F-distribution
{(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\! | cdf = I_{\frac{d_1 x}{d_1 x + d_2}} \left(\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) | mean = \frac{d_2}{d_2-2}\! for d''2 > 2| median =| mode = \frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\! for ''d''1 > 2| variance = \frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\! for ''d''2 > 4| skewness = \frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\! for ''d''2 > 6| kurtosis =''see text| entropy =| mgf =''does not exist, raw moments defined in text and in | char =''see text|}} In probability theory and statistics, the F-distribution is a continuous probability distribution. NIST (2006). Engineering Statistics Handbook - F Distribution It is also known as Snedecor's F distribution or the Fisher-Snedecor distribution (after R.A. Fisher and George W. Snedecor). The F-distribution arises frequently as the null distribution of a test statistic, most notably in the analysis of variance; see F-test. Definition If a random variable X'' has an F-distribution with parameters ''d''1 and ''d''2, we write ''X ~ F(d''1, ''d''2). Then the probability density function for ''X is given by : \begin{align} f(x; d_1,d_2) &= \frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \\ &=\frac{1}{\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{\frac{d_1}{2}} x^{\frac{d_1}{2} - 1} \left(1+\frac{d_1}{d_2}\,x\right)^{-\frac{d_1+d_2}{2}} \end{align} for real x'' ≥ 0. Here \mathrm{B} is the beta function. In many applications, the parameters ''d''1 and ''d''2 are positive integers, but the distribution is well-defined for positive real values of these parameters. The cumulative distribution function is : F(x; d_1,d_2)=I_{\frac{d_1 x}{d_1 x + d_2}}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) , where ''I is the regularized incomplete beta function. The expectation, variance, and other details about the F(d''1, ''d''2) are given in the sidebox; for ''d''2 > 8, the excess kurtosis is : \gamma_2 = 12\frac{d_1(5d_2-22)(d_1+d_2-2)+(d_2-4)(d_2-2)^2}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)} . The ''k-th moment of an F(d''1, ''d''2) distribution exists and is finite only when 2''k < d''2 and it is equal to : \mu _{X}(k) =\left( \frac{d_{2}}{d_{1}}\right)^{k}\frac{\Gamma \left(\tfrac{d_1}{2}+k\right) }{\Gamma \left(\tfrac{d_1}{2}\right) }\frac{\Gamma \left(\tfrac{d_2}{2}-k\right) }{\Gamma \left( \tfrac{d_2}{2}\right) } The ''F-distribution is a particular parametrization of the beta prime distribution, which is also called the beta distribution of the second kind. The characteristic function is listed incorrectly in many standard references (e.g., ). The correct expression Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261-264 is : \varphi^F_{d_1, d_2}(s) = \frac{\Gamma(\frac{d_1+d_2}{2})}{\Gamma(\tfrac{d_2}{2})} U \! \left(\frac{d_1}{2},1-\frac{d_2}{2},-\frac{d_2}{d_1} \imath s \right) where U''(''a, b'', ''z) is the confluent hypergeometric function of the second kind. Characterization A random variate of the F-distribution with parameters d''1 and ''d''2 arises as the ratio of two appropriately scaled chi-squared variates:M.H. DeGroot (1986), ''Probability and Statistics (2nd Ed), Addison-Wesley. ISBN 0-201-11366-X, p. 500 : X = \frac{U_1/d_1}{U_2/d_2} where *''U''1 and U''2 have chi-squared distributions with ''d''1 and ''d''2 degrees of freedom respectively, and *''U''1 and ''U''2 are independent. In instances where the F-distribution is used, for example in the analysis of variance, independence of ''U''1 and ''U''2 might be demonstrated by applying Cochran's theorem. Equivalently, the random variable of the F-distribution may also be written : X = \frac{s_1^2}{\sigma_1^2} \;/\; \frac{s_2^2}{\sigma_2^2} where ''s''12 and ''s''22 are the sums of squares ''S''12 and ''S''22 from two normal processes with variances σ12 and σ22 divided by the corresponding number of χ2 degrees of freedom, ''d''1 and ''d''2 respectively. In a Frequentist context, a scaled F-distribution therefore gives the probability ''p(s''12/''s''22 | σ12, σ22), with the F distribution itself, without any scaling, applying where σ12 is being taken equal to σ22. This is the context in which the F-distribution most generally appears in F-tests: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis. The quantity ''X has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of σ12 and σ22.G.E.P. Box and G.C. Tiao (1973), Bayesian Inference in Statistical Analysis, Addison-Wesley. p.110 In this context, a scaled F-distribution thus gives the posterior probability p''(σ22/σ12|''s''12, ''s''22), where now the observed sums ''s''12 and ''s''22 are what are taken as known. Generalization A generalization of the (central) F-distribution is the noncentral F-distribution. Related distributions and properties *If X \sim \chi^2_{d_1} and Y \sim \chi^2_{d_2} are independent, then \frac{X / d_1}{Y / d_2} \sim \mathrm{F}(d_1, d_2) *If X \sim \operatorname{Beta}(d_1/2,d_2/2) (Beta distribution) then \frac{d_2 X}{d_1(1-X)} \sim \operatorname{F}(d_1,d_2) *Equivalently, if ''X ~ F(d''1, ''d''2), then \frac{d_1 X/d_2}{1+d_1 X/d_2} \sim \operatorname{Beta}(d_1/2,d_2/2) . *If ''X ~ F(d''1, ''d''2) then Y = \lim_{d_2 \to \infty} d_1 X has the chi-squared distribution \chi^2_{d_1} *F(''d''1, ''d''2) is equivalent to the scaled Hotelling's T-squared distribution \frac{d_2}{d_1(d_1+d_2-1)} \operatorname{T}^2 (d_1, d_1 +d_2-1) . *If ''X ~ F(d''1, ''d''2) then ''X−1 ~ F(d''2, ''d''1). *If ''X ~ [[Student's t-distribution|t(n'')]] then :: X^{2} \sim \operatorname{F}(1, n) :: X^{-2} \sim \operatorname{F}(n, 1) *F-distribution is a special case of type 6 Pearson distribution *If ''X and Y'' are independent, with ''X, Y'' ~ [[Laplace distribution|Laplace(μ, ''b)]] then :: \tfrac \sim \operatorname{F}(2,2) *If X'' ~ F(''n, m'') then \tfrac{\log{X}}{2} \sim \operatorname{FisherZ}(n,m) (Fisher's z-distribution) *The noncentral F-distribution simplifies to the F-distribution if λ = 0. *The doubly noncentral F-distribution simplifies to the F-distribution if \lambda_1 = \lambda_2 = 0 *If \operatorname{Q}_X(p) is the quantile ''p for X'' ~ F(''d''1, ''d''2) and \operatorname{Q}_Y(1-p) is the quantile 1−''p for Y'' ~ F(''d''2, ''d''1), then :: \operatorname{Q}_X(p)=\frac{1}{\operatorname{Q}_Y(1-p)} . See also * Chi-squared distribution * Chow test * Gamma distribution * Hotelling's T-squared distribution * Student's t-distribution * Wilks' lambda distribution * Wishart distribution References Further reading Key texts Books Papers Additional material Books Papers External links *[http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm Table of critical values of the ''F-distribution] *Online significance testing with the F-distribution *Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution Category:Continuous distributions